Definition:Quotient Group/Motivation
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Motivation for Definition of Quotient Group
In Kernel is Normal Subgroup of Domain it was shown that the kernel of a group homomorphism is a normal subgroup of its domain.
In that result it has been shown that every normal subgroup is a kernel of at least one group homomorphism of the group of which it is the subgroup.
We see that when a subgroup is normal, its cosets make a group using the group operation defined as in this result.
However, it is not possible to make the left or right cosets of a non-normal subgroup into a group using the same sort of group operation.
Otherwise there would be a group homomorphism with that subgroup as the kernel, and we have seen that this can not be done unless the subgroup is normal.
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Quotient Groups: Theorem $3$